Lacerations and any injury
of an extremity such as the hand are more concerning than most little injuries to other body parts because of the potential for injury to the nerves and blood vessels that supply them. For this reason, it is always best to see a physician if there is any doubt about the laceration itself, and immediately see a doctor
if you have any inability to move your hand or fingers completely or there are any sensory deficits (you can feel touch). For extremely simple lacerations that are very superficial, there are several things that will help the wound to heal. Adequate blood flow is one of the most important, and so any peripheral vascular disease would slow down the process. Diabetes or other autoimmune diseases (or even taking a prescription steroid regularly) would also slow down the healing process. Advanced age can also be a factor in and of itself. Finally, too many bacteria can cause a wound to heal more slowly, which is why different people recommend different topical antibiotics, as you have mentioned in your question. Iodine, alcohol, and antibiotics that you take by mouth or intravenously all have the same basic effect: they decrease the amount of bacteria in the wound, allowing normal healing to take place. For most simple wounds, a topical antibiotic ointment that you can get over the counter, such as bacitracin, is completely adequate. Look out for signs of infection, such as increased redness, green or yellow discharge, or fever/chills, and see a physician if there is any doubt.